A directly modulated laser may be used as an optical transmitter that transmits light at a given wavelength. The power (i.e., amplitude) of the laser light is modulated by corresponding modulation of the current used to drive the laser. For example, the optical transmitter may be modulated to carry a wide-band RF signal. In this case, the electrical current that drives or pumps the laser is modulated with the wide-band RF signal.
The use of a directly-modulated laser to carry a wide-band RF signal may result in distortion due to the multiple carrier frequencies of the multichannel RF signal modulating the laser and/or the harmonics produced by the non-linear nature of the laser device. Intermodulation distortion may be produced when two or more signals (e.g., 2 or more carriers) mix together to form distortion products. Discrete distortion may be produced from only one carrier. Distortion may include even-order distortion and odd-order distortion. In a CATV system, the most significant types of even-order and odd-order distortion products are second-order distortion products and third-order distortion products, respectively. Second-order intermodulation (IM2) distortion products may include, for example, intermodulation products formed by combining signals at frequencies A and B to produce new signals at the combined frequencies, such as A±B. Third-order intermodulation (IM3) distortion products may include, for example, intermodulation products formed by combining signals at frequencies A, B, and C to produce new signals at frequencies A±B±C and 2A±B.
In a CATV system, there are a multitude of carriers spaced equally in frequency, which may produce numerous intermodulation distortion products that lie at the same frequency. The sum of second-order intermodulation products that are present at a particular frequency is commonly referred to as composite second order (CSO) distortion. In a CATV system, the equal spacing of the carriers may also cause multiple third-order intermodulation products to line up at the same frequency and directly on top of the carrier frequency. The sum of these third-order intermodulation products that are present in a particular channel is commonly referred to as composite triple beat (CTB) distortion.
The non-linearities of a time-independent non-linear element, such as an amplifier, may be modeled as Taylor series expansions or power series expansions of an input signal. For example, the output y of a non-linear amplifier may be described as a Taylor series expansion of an input x:y(x)=Co+C1x+C2x2+C3x3+C4x4+ . . . Ckxk  Eq. 1where C0, C1, C2, C3, C4, . . . Ck are constants representative of the behavior of the non-linear amplifier. The order within the series is determined by the highest power of x in the expansion. The even order (x2n where n=1, 2, 3 . . . ) terms in the series (e.g., C2x2, C2x4, C2x6, . . . ) represent even order distortion and the odd order (x2n+1 where n=1, 2, 3 . . . ) terms in the series (e.g., C2x3, C2x5, C2x7, . . . ) represent odd order distortion. For example, C2x2 is the second-order term and represents distortion from the first of the even order terms and C3x3 is the third-order term and represents distortion from the first of the odd order terms. When the input x is an RF input, both x and y are time-varying quantities. With an input having two angular frequencies (ω1 and ω2) represented as x=a sin(ω1t)+b sin(ω2t), the second order term C2x2 creates second order distortion products at frequencies 2ω1, 2ω2, ω1−ω2, and ω1+ω2. Because the non-linear element in this case is time independent, the magnitude and phase of these distortion products are not dependent upon the modulation frequency.
When the non-linear element also has time dependence, such as for lasers, the Taylor series is expanded to include the time dependent terms as follows:
                              y          ⁡                      (                          x              ⁡                              (                t                )                                      )                          =                              C            00                    +                                    C              01                        ⁢            x                    +                                    C              02                        ⁢                          x              2                                +                                    C              03                        ⁢                          x              3                                +                      …            ⁢                                                  ⁢                          C                              0                ⁢                k                                      ⁢                          x              k                                +                                    C              11                        ⁢                                          ⅆ                x                                            ⅆ                t                                              +                                    C              12                        ⁢            x            ⁢                                          ⅆ                x                                            ⅆ                t                                              +                                    C              13                        ⁢                          x              2                        ⁢                                          ⅆ                x                                            ⅆ                t                                              +                      …            ⁢                                                  ⁢                          C                              1                ⁢                k                                      ⁢                          x                              k                -                1                                      ⁢                                          ⅆ                x                                            ⅆ                t                                              +                                    C              21                        ⁢                                                            ⅆ                  2                                ⁢                x                                            ⅆ                                  t                  2                                                              +                                    C              22                        ⁢            x            ⁢                                                            ⅆ                  2                                ⁢                x                                            ⅆ                                  t                  2                                                              +                                    C              23                        ⁢                          x              2                        ⁢                                                            ⅆ                  2                                ⁢                x                                            ⅆ                                  t                  2                                                              +                      …            ⁢                                                  ⁢                          C                              2                ⁢                k                                      ⁢                          x                              k                -                1                                      ⁢                                                            ⅆ                  2                                ⁢                x                                            ⅆ                                  t                  2                                                              +                                    C                              n                ⁢                                                                  ⁢                1                                      ⁢                                                            ⅆ                  n                                ⁢                x                                            ⅆ                                  t                  n                                                              +                                    C                              n                ⁢                                                                  ⁢                2                                      ⁢            x            ⁢                                                            ⅆ                  n                                ⁢                x                                            ⅆ                                  t                  n                                                              +                                    C                              n                ⁢                                                                  ⁢                3                                      ⁢                          x              2                        ⁢                                                            ⅆ                  n                                ⁢                x                                            ⅆ                                  t                  n                                                              +                      …            ⁢                                                  ⁢                          C              nk                        ⁢                          x                              k                -                1                                      ⁢                                                            ⅆ                  n                                ⁢                x                                            ⅆ                                  t                  n                                                                                        Eq        .                                  ⁢        2            
When an input having two angular frequencies (ω1 and ω2) represented as x=a sin(ω1t)+b sin(ω2t) is applied to the above time dependent non-linear element, the second order distortion at frequencies 2ω1, 2ω2, ω1−ω2, and ω1+ω2 will have an amplitude and phase that is dependent on frequency. For the 2ω1 term, the dependence may be represented as follows:
                              y                      2            ⁢                          ω              1                                      =                                            a              2                        2                    [                                                    C                02                            ⁢                              cos                ⁡                                  (                                      2                    ⁢                                          ω                      1                                        ⁢                    t                                    )                                                      +                                          C                12                            ⁢                              ω                1                            ⁢                              sin                ⁡                                  (                                      2                    ⁢                                          ω                      1                                        ⁢                    t                                    )                                                      -                                          C                22                            ⁢                              ω                1                2                            ⁢                              cos                ⁡                                  (                                      2                    ⁢                                          ω                      1                                        ⁢                    t                                    )                                                      +            …                    ⁢                                          ]                                    Eq        .                                  ⁢        3            
The first term in the above series represents the frequency independent term. The remaining terms represent frequency dependent terms that are a result of the time dependence upon distortion. A similar dependence can be found for other second order distortion products.
For any given non-linear element, such as a laser, the magnitude and sign of the coefficients of the time dependent Taylor series expansions are often unknown. Furthermore, the magnitude and sign of the coefficients can change with parameters such as laser power or temperature. When multiple non-linear elements are present in a system, such as the case for hybrid fiber coax transmission systems using direct modulated lasers, the coefficients of the Taylor series expansion describing the system will be related to the sum of the respective coefficients describing the non-linear elements within the system. Other non-linear elements in a hybrid fiber coax transmission system could be, for example, the fiber used to transmit the optical signal. The result of summing these coefficients is that not only is the magnitude of the system coefficients often unknown, so is the sign. Also both magnitude and sign of the system coefficient can change with system parameters.
Several techniques have been proposed or employed to compensate for distortion by injecting distortion of equal magnitude but opposite phase to the distortion produced by the laser device. For example, a predistortion circuit may be employed to predistort the RF signal being applied to modulate the laser. One such predistortion circuit includes split signal paths—a main or primary signal path and a secondary signal path. A small sample of the RF input is tapped off the main signal path and a distortion generator in the secondary signal path generates distortion (i.e., predistortion). The predistortion is then recombined with the RF signal on the main signal path such that the predistortion is of equal magnitude but opposite sign to the laser-induced distortion.
These predistortion circuits have been proposed or employed using frequency independent magnitude adjustments in the secondary path and even magnitude-phase tilt filters to account for the frequency dependent effects. However, such existing predistortion circuits may not be effective to compensate for element and/or system distortion both initially and/or during operation and/or with changes in system parameters. One of the reasons may be the inability to change the phase of the secondary path(s) by 180° to account for the possibility that the coefficients or sum of coefficients of the time dependent Taylor series expansion will change signs.